#### Holography of the BTZ black hole, inside and out

Anton de la Fuente
0
Raman Sundrum
0
0
Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park
,
20782 MD, U.S.A
We propose a 1 + 1 dimensional CFT dual structure for quantum gravity and matter on the extended 2 + 1 dimensional BTZ black hole, realized as a quotient of the Poincare patch of AdS3. The quotient spacetime includes regions beyond the singularity, whiskers, containing timelike and lightlike closed curves, which at first sight seem unphysical. The spacetime includes the usual AdS-asymptotic boundaries outside the horizons as well as boundary components inside the whiskers. We show that local boundary correlators with some endpoints in the whisker regions: (i) are a protected class of amplitudes, dominated by effective field theory even when the associated Witten diagrams appear to traverse the singularity, (ii) describe well-defined diffeomorphism-invariant quantum gravity amplitudes in BTZ, (iii) sharply probe some of the physics inside the horizon but outside the singularity, and (iv) are equivalent to correlators of specific non-local CFT operators in the standard thermofield entangled state of two CFTs. In this sense, the whisker regions can be considered as purely auxiliary spacetimes in which these useful non-local CFT correlators can be rendered as local boundary correlators, and their diagnostic value more readily understood. Our results follow by first performing a novel reanalysis of the Rindler view of standard AdS/CFT duality on the Poincare patch of AdS, followed by exploiting the simple quotient structure of BTZ which turns the Rindler horizon into the BTZ black hole horizon. While most of our checks are within gravitational effective field theory, we arrive at a fully non-perturbative CFT proposal to probe the UV-sensitive approach to the singularity. ArXiv ePrint: 1307.7738 Open Access, c The Authors. Article funded by SCOAP3.
1 Introduction
2.1 Diffeomorphism invariance in non-perturbative formulation 2.2 Strategy for BTZ
2.3 Through the singularity: the whisker regions
2.4 Space time inside the horizon
2.5 Comparing whiskers and Euclidean space as auxiliary spacetimes
2.6 Whisker correlators as generalizing in-in correlators
2.7 Studying the singularity
2.8 Relation to the literature
2.9 Organization of paper
BTZ as quotient of AdSPoincare
The extended BTZ boundary and challenges for the CFT dual
4.1 BTZ boundary as disconnected cylinders
4.2 Connected view of BTZ
Boundary correlators and the singularity
5.1 Approaching singularity from outside
5.2 Flawed attempt to scatter through singularity
5.3 Approaching singularity from inside
5.4 Proper account of scattering through singularity
5.5 Matching to CFT on BTZ
6 Space time inside the horizon
6.1 x t in free CFT
6.2 x t in Milne wedges and reconnecting to Rindler wedges
6.3 CFT on BTZ as a trace
CFT dual in thermofield form
7.1 Special case of purely Rindler wedge sources
7.2 General case of arbitrary sources
8 rS = : Rindler AdS/CFT
8.1 Special case of purely Rindler wedge sources
8.2 Comparison with dual CFT
8.3 General case of arbitrary sources
8.4 Diagrammatic analysis of thermofield formulation
8.5 Testing boundary localized correlators (in all regions)
8.6 Testing general bulk correlators 1 4 4
9.2 Local boundary correlators: EFT dominance and scattering behind the horizon 39
9.3 Method of images applied to Rindler AdS/CFT 40
9.4 Connecting to CFT dual on BTZ 41
9.5 Deeper reason for insensitivity to singularity 43 10 Sensing near-singularity physics 11 Comments and conclusions 45
Nearly a century after the discovery of the Schwarzschild metric,
rS
ds2 = 1 r
dt2 1 dr2rrS r2 d2 + sin2 d2 ,
black holes remain a source of mystery and fascination. In theoretical physics, they provide
key insights for our most ambitious attempts to unify gravity, relativity and quantum
mechanics. Viewed from the outside as robust endpoints of gravitational collapse, and
decaying subsequently via Hawking radiation, black holes pose the information paradox.
Falling inside, the roles of time, , and space, r, apparently trade places, the horizon now
encompassing a universe within, with the future singularity its big crunch. Understanding
these dramatic phenomena seems tantalizingly close to our grasp, just beyond the horizon, a
region comprised of familiar, smooth patches of spacetime. And yet, the local simplicity of
the horizon belies its global subtlety, which still lacks an explicit inside/outside description
within a fundamental framework for quantum gravity (as exemplified by the recent firewall
paradox [13]1) regarding evaporating black holes. Nevertheless, powerful ideas and results
in holography [5, 6], complementarity [7], string theory and AdS/CFT duality [810]
(reviewed in [1113]), have combined with gravitational effective field theory (EFT) to give
us a much clearer picture of the central issues (reviewed in [14, 15]).
In such a situation, it is natural to look for an Ising model, a special case that enjoys
so many technical advantages that we can hope to solve it exactly, and whose solution would
test and crystalize tentative grand principles, and brings new ones to the fore. For this
purpose, the 2 + 1-dimensional BTZ black hole [16, 17] is, in many ways, an ideal candidate.
The BTZ geometry solves Einsteins Equations with negative cosmological constant in 2 + 1
dimensions, and is given in Schwarzschild coordinates by,
1See also [4] for a prediction similar to firewalls from different assumptions.
not that dissimilar from (1.1). The geometry asymptotes for large r to that of global anti-de
Sitter spacetime, AdS3 global, with radius of curvature RAdS and AdS boundary at r = .
The horizon is at the Schwarzschild radius, r = rS. It is the simplest of the large AdS
Schwarzschild black holes, eternal in that they do not decay via Hawking radiation, but
rather are in equilibrium with it [18]. It retains many of the key interesting features of
black holes in general. In what follows it will be more convenient to rescale coordinates,
and to switch to RAdS 1 units, so the metric becomes
The horizon is now at r = 1.
Although pure 2 + 1-dimensional general relativity does not contain propagating
gravitons, it does have gravitational fluctuations and backreactions, and coupled to propagating
matter the EFT is non-renormalizable as in higher dimensions (in fact, it may be a
compactification of higher dimensions, and contain propagating Kaluza-Klein gravitons), requiring
UV completion. It also shares with higher-dimensional eternal AdS Schwarzschild black
holes, the central consequence of AdS/CFT duality: as an object inside AdSglobal the black
hole inherits a holographic dual in terms of a hot conformal field theory (CFT) (for BTZ,
a 1 + 1 CFT on a spatial circle), the CFT temperature being dual to the BTZ Hawking
temperature. More precisely [19] (see also the earlier steps and insights of [2022]), the
duality is framed in terms of the Kruskal extension of BTZ,
ds2 =
The horizon, singularity and AdS boundaries are now as follows:
The Penrose diagram of this spacetime is shown in figure 1. BTZ is seen to interpolate
between two distinct asymptotically-AdSglobal boundary regions. The holographic dual is
then given by two CFTs, dynamically decoupled, but in a state of thermofield [2329]
entanglement,
The entangled state is dual to the Hartle-Hawking choice of vacuum [30] for the BTZ
black hole.
There remains the puzzle of detailing just how this CFT description incorporates
processes inside the BTZ horizon. We know that in asymptotic AdS spacetimes, the set
Figure 1. The Penrose diagram of the extended BTZ black hole spacetime. The vertical lines
represent the boundaries of two asymptotically AdS regions.
of local boundary correlators gives a beautiful diffeomorphism-invariant quantum
gravity description of scattering which generalizes the S-matrix construction of asymptotic
Minkowski spacetimes, and, in the sense described in [31], is even richer in structure.
Furthermore, these boundary correlators have a non-perturbative and UV-complete description
in terms of correlators of local CFT operators living on the AdS boundary, AdS. But
in AdS-Schwarzchild spacetimes like BTZ it is not apparent what CFT questions give a
diffeomorphism-invariant and non-perturbative description of scattering inside the horizon:
one can send in wavepackets from outside the horizon aimed to scatter within, but the
products of any scattering must causally end up at the singularity rather than returning to the
exterior AdS boundaries. While one can connect Witten diagrams from interaction points
in the interior of the (future) horizon to the boundaries shown in figure 1, these connections
cannot sharply capture the fate of such interactions since they are at best spacelike.
This does not mean that the interior of the horizon is out of bounds to the CFT
description. In a sense, what is required is a set of out states consisting of approximately
decoupled bulk particles located on a spacelike hypersurface before the (future) singularity,
with which one can compute the overlap with the state resulting from the scattering process.
Even in (the simpler) AdS spacetime, particles inside the bulk are described by non-local
disturbances of the CFT, so one can anticipate that any holographic description of scattering
inside the horizon will necessarily involve correlators of non-local CFT operators. But
specifically which non-local CFT operators correspond to the simplest basis of out states,
so that their correlators (with other CFT operators) provide a sharp diagnostic of scattering
inside the horizon? In this paper, we identify such non-local CFT operators and demonstrate
that they correspond to the intuitive notion of scattering inside the horizon. Our proposal
is precisely and non-perturbatively framed. We test it by applying it to scattering inside
the horizon but far from the singularity where, at short distances RAdS, the behavior is
very much like scattering outside the horizon or in flat spacetime, and so we know what
to expect. We then show how to apply our proposal to probe the more mysterious regime
near the singularity, where EFT breaks down and even perturbative string theory may be
blind to important non-perturbative effects (see for example, [19]). Since the interior of the
horizon is a cosmological spacetime, finding the non-local CFT operators can be thought of
as giving the holographic description of a quantum cosmology with singularity, a signficant
step beyond the more familiar holography of static AdS.
Overview and organization 2
Diffeomorphism invariance in non-perturbative formulation
The issue of diffeomorphism invariance, and the challenge it poses for a description of the
interior of the horizon, may seem unfamiliar to those who routinely use local field operators
to sharply describe processes in the real world (which of course includes quantum gravity in
some form). This would naively suggest that in the BTZ context we should use local bulk
operators acting on the Hartle-Hawking state to create in/out states inside the horizon, and
then translate these operators to (non-local) operators of the CFT. However, fundamentally
all local fields (composite or elementary) violate the diffeomorphism gauge symmetry of
quantum gravity (their spacetime argument at least is not generally coordinate-invariant),
just as the local electron and gauge fields violate gauge invariance in QED. Of course, we
are used to using gauge non-invariant local operators within a gauge-fixed formalism, but
these are, in essence, non-local constructions in the gauge-invariant data. For example
in electromagnetism, the gauge-invariant data (in Minkowski spacetime) are provided by
specifying some field strength, F (x), subject to the Bianchi identity, F (x) = 0.
This uniquely determines a local gauge potential, A(x) : A A = F , once we
stipulate some gauge-fixing condition (and behavior at infinity), such as
A(x) is thereby a non-local functional of F (y). In this way, gauge-fixing is seen as a
method for giving non-local gauge-invariant operators a superficially local (and useful) form.
In gravity, the gauge-fixing approach is useful for perturbatively small fluctuations of the
metric, but not when there are violent fluctuations of the metric (or when the notion of
spacetime geometry itself breaks down). And yet it is precisely large fluctuations of the
metric that we are interested in when we are concerned with non-perturbative effects (in
GNewton) saving us from information loss (see discussion in [19]), or in the approach to
the singularity. Therefore, in the non-perturbative framing of our proposal we avoid the
intermediate step of gauge-fixed local bulk fields, instead exploiting the greater simplicity
of BTZ over other black holes to directly identify the (diffeomorphism-invariant) CFT
observables.
Nevertheless, it is useful to see how our approach reduces to gauge-fixed EFT of bulk
fields, when that is valid, and this also provides an arena for testing the proposal. To
this end, we will show that correlators of local field operators inside the horizon can be
re-expressed as correlators of non-local EFT observables outside the horizon (in principle
accessible to an outside observer). Even though this dictionary is between gravitational
EFT descriptions, the translating operation is non-perturbative in form. It resonates
with the ideas of complementarity [7], where the interior of the horizon is not independent
of the exterior, but rather a very different probe of it.
Strategy for BTZ
BTZ is particularly well-suited to address the above issues for two reasons. First, the
enhanced conformal symmetry of 1 + 1-dimensional CFTs over higher dimensions provides
(b) Boundary operators on the left and
right Rindler wedges are spacelike
separated from the scattering event.
us with a better understanding of their properties. The second reason is that BTZ can be
realized as a quotient of AdS spacetime itself, by identifying points related by a discrete
AdS isometry [16, 17]. At the technical level, BTZ Green functions can be easily obtained
from the highly symmetric AdS Green functions using the method of images [32, 33]. Most
importantly, the BTZ horizon emerges as the quotient of a mere Rindler horizon, as
would be seen by a class of accelerating observers in AdS [34]. (See [35] for a related
discussion, and [36] for a higher-dimensional discussion.) The Rindler view of AdS, the BTZ
black string, is given by (1.2) and (1.4) again, but now with non-compact (, +),
rS . Our approach is based on a novel reanalysis of Rindler AdS/CFT [19, 36], in a
manner that can then be straightforwardly quotiented to the BTZ case of interest.
The central issue from the Rindler view can be seen in figure 2a, depicting the Poincare
patch of AdS, where the intersecting planes are the Rindler horizons, light rays travel at 45
degrees to the vertical time axis, the boundary is at z = 0, and
x t x
are boundary (1 + 1 Minkowski) lightcone coordinates. Two particles are seen to enter the
future horizon, scatter inside, and then the resultant particle lines measured by local
boundary correlators ending in the boundary region inside the horizon. Such correlators can
sharply diagnose the results of the scattering because the endpoints are causally connected
to the scattering point (or region). In Minkowski CFT these endpoints correspond to local
operators in the Milne wedge inside the Rindler horizon. However, the dual Rindler CFT
picture corresponds to two CFTs living only in the left and right regions outside the
horizon (entangled with each other in the thermofield state), so that correlators of local
CFT operators correspond to boundary correlators only ending in the boundary regions
outside the horizon. As seen in figure 2b, such local Rindler CFT correlators correspond
to boundary correlators with endpoints at best spacelike separated from the scattering
point, not useful for a sharp diagnosis of the scattering (as we already saw from the Penrose
diagram of figure 1).
However, the desired local operators of the Minkowski CFT (as opposed to the Rindler
CFTs) inside the horizon have the form,
O(t, x) eiHMinktO(0, x)eiHMinkt,
where the operator at t = 0 is now within the Rindler region and equivalent to a local Rindler
CFT operator. The Minkowski CFT Hamiltonian HMink is also some operator on the tensor
product of the Hilbert spaces of the two Rindler CFTs (= Hilbert space of the Minkowski
CFT, as is apparent at t = 0), so O(t, x) must also be some operator of the Rindler
CFTs. But because HMink 6= HRindler, O(t, x) is not simply a local Heisenberg operator of
the Rindler CFTs, but rather non-local from the Rindler perspective. We conclude that
non-local correlators of the Rindler CFTs are able to sharply capture scattering inside
the Rindler horizon, the same way that local correlators of the Minkowski CFT ending
inside the horizon do. The problem in taking the BTZ quotient of this nice story is that
the quotient of HMink does not exist: the associated t-translation isometry is broken by
quotienting.
An important result of ours is to reproduce the correlators of (2.3), which sharply
capture scattering inside the Rindler horizon, with a new set of non-local Rindler CFT
operators,
Onon-local e 2 (HRindlerPRindler)Olocale 2 (HRindlerPRindler),
constructed from local Rindler CFT operators Olocal and the Rindler Hamiltonian and
momentum, HRindler, PRindler. Note that we are not equating these new non-local operators
with those of (2.3); they will have different matrix elements within generic states. We
only show that they have the same matrix elements in a fixed, special state, namely the
thermofield state of the two Rindler CFTs, namely |i for rS = . This suffices to
capture scattering inside the Rindler horizon. But unlike (2.3), the new operators are
straightforwardly quotiented to the CFT dual of BTZ. Indeed, this quotient is simply the
compactification of the spatial Rindler direction, so that PRindler becomes the conserved
angular momentum of the thermofield CFTs on a spatial circle, and HRindler becomes their
Hamiltonian. We will show that the resulting non-local operators in the thermofield CFTs
have correlators which provide some sharp probes of scattering inside the BTZ horizon.
This conclusion is certainly subtle and delicate, as illustrated in figure 3. After
quotienting AdS to BTZ, the lightcones in figure 3 become the future and past singularities. So it
would appear that the quotient construction of correlators to see the scattering inside the
horizon will correspond to the analog of figure 2a in BTZ, a diagram that necessarily
traverses the singularity. This raises the question of whether the quotienting procedure outlined
above is straightforward and trustworthy. Indeed we claim it is, but to double-check this
requires studying the singularity more closely, and Feynman diagrammatics in its vicinity.
Through the singularity: the whisker regions
Purely at the level of the spacetime geometry (before any dynamics is considered), the
quotient construction gives BTZ a perfectly smooth passage (with finite curvature) through
the singularity, the quotient of the lightcones of figure 3 (r = 0 or uv = 1 in Schwarzchild
and Kruskal coordinates respectively). However, after the quotient the regions inside these
lightcones contain closed timelike curves, dubbed whiskers in [37]. (In the Rindler limit,
rS , these closed curves become infinitely long and the whiskers revert to just ordinary
parts of AdS.) The smoothness of the quotient geometry is also deceptive, and the singularity
well deserves its name once one makes any attempt to physically probe it. After quotienting
the lightcones of figure 3, they are comprised of closed lightlike curves where even small
(quantum) fluctuations [38, 39] can backreact divergently with divergent curvatures, and
general considerations imply the breakdown of ordinary (effective) field theory [40]. See [41]
for a concise review of these general considerations. Similar singularities have also been
studied in the context of string theory. Attempts to scatter through the singularity in string
theory failed to obtain well-defined amplitudes (see [42] for a concise review and original
references). Ref. [43] found that stringy effects involving the twisted sector smoothed out
the large backreactions, but so as to isolate the spacetime regions outside the singularity
from the whisker (and other) regions beyond the singularity. In any case, much of the
literatures suggests that the whisker regions are both wildly unphysical and inaccessible
because of the singularity. This seems at odds with our claim that diagrams ending in the
whisker regions are the dual of the non-local CFT correlators described above, and that
these capture scattering inside the horizon.
However, we will show that local boundary correlators with some endpoints in the
whisker regions are in fact well-defined, and a protected sub-class are dominated within
EFT, parametrically insensitive to what happens very close to the singularity, even when the
associated (Witten) diagrams traverse the singularity. This protected sub-class is specified
by first noting that the maximal extension of the BTZ black hole spacetime is given by [17]
where is a quotient discrete isometry group of AdS. An intermediate extension of the black
hole spacetime is then given by replacing AdSglobal with just the Poincare patch, AdSPoincare.
This still includes the entire Kruskal extension of BTZ as well as two whisker regions,
The protected class of boundary correlators is precisely the set confined to AdSPoincare/,
rather than all of AdSglobal/. For this reason, we confine ourselves in this paper to
AdSPoincare/, and simply identify it in what follows as the BTZ spacetime. We will
return in future work to a treatment of the boundary correlators of the maximally extended
BTZ spacetime given by AdSglobal/ [44].
Technically, in the AdSPoincare/ realization of BTZ, naive divergences appear when
Witten diagram interaction vertices approach the singularity, but are rendered finite by (a)
using and tracking the correct i prescription in BTZ propagators, following from AdS
propagators by the method of images, and (b) including the whiskers in the integration
region for interaction vertices. Roughly,
< ,
where r is the Schwarzchild radial coordinate for r > 0 and a related coordinate inside
the whisker region for r < 0. Clearly, the finiteness of such expressions as 0 requires
integrating into the whisker region, r < 0. (Similar cancellations were noted in [45]).
More strongly, we will show that many of the BTZ local boundary correlators are
well-approximated by the analogous diagrams in (unquotiented) AdSPoincare itself, where
the interpretation in terms of scattering behind the (Rindler) horizon is unambiguous. This
is the basis of our claim that we have found a class of correlators sensitive to scattering
behind the BTZ horizon.
Space time inside the horizon
Despite these good features, correlators in regions with timelike closed curves seem at
odds with a physical interpretation and connection to the standard thermofield CFT dual.
Relatedly, it is puzzling why we are lucky enough that the associated Witten diagrams
should be insensitive to what is happening close to the singularity. We show that these
correlators can be put into a more canonical form by performing a well-defined space
time transformation which takes local operators inside the horizon into non-local
operators outside the horizon (and thereby make them accessible to external observers).
This transformation is particularly plausible in the dual 1 + 1 CFT where the causal
(lightcone) structure is symmetric between space and time, and indeed we show that the
transformation can be viewed as a kind of improper conformal transformation. It is this
transformation that ultimately leads to the non-local operators arising from local ones,
seen in (2.4). Such a symmetry seems much less manifest from the AdS perspective where
there is no such isometry, but we prove that it indeed exists as an unexpected symmetry of
boundary correlators, by a careful Witten-diagrammatic analysis.
In more detail, the transformation is also accompanied by complex phases that are
necessary for ensuring relativistic causality constraints in correlators, naively threatened
because spacelike timelike.) We thereby interpret our results as having found (i)
nonlocal CFT operators that simply describe scattering inside the BTZ horizon (but outside
the singularity), and (ii) an auxiliary but bizarre spacetime extension of the BTZ black hole,
whiskers, in which these non-local CFT operators are rendered as local operators, and
in which some of their properties become more transparent. Whether or not one thereby
considers the whiskers to be physical regions is left to the reader.
Comparing whiskers and Euclidean space as auxiliary spacetimes
The notion of an auxiliary spacetime grafted onto the physical spacetime, where one uses
path integrals and operators in the former to implant certain types of wavefunctionals in
the latter, is already familiar when the auxiliary spacetime is Euclidean. For example, such
constructions are used to create the Hartle-Hawking wavefunctional [30] or its perturbations
in the physical spacetime, and can have a non-perturbative CFT dual [19]. Indeed, they
too can be used to create quite general bulk states in the interior of BTZ, in principle
including the kind of out states for scattering that we seek. However, the simple Euclidean
constructions yield physical states at the point of time symmetry, u + v = 0 (or = 0).
We would need to evolve these states to late time and take superpositions in order to find
out states that consist of several approximately free bulk particles. The problem then is
that identifying such superpositions is equivalent to solving the scattering dynamics itself!
By contrast, the virtue of our Lorentzian auxiliary spacetime whisker is that it allows
us to create simple out states with simply defined operators. In this way, we can pose
explicit (non-local) CFT correlators which capture the fate of scattering inside the horizon.
A well-programmed CFT computer would then output the answers to such questions
without first requiring equally difficult computations as input.
Whisker correlators as generalizing in-in correlators
It is not simply fortuitous that Witten diagrams are insensitive to the singularity, even
with some endpoints on the boundary of the whisker regions. Rather, we will show that the
approach to the singularity in the bulk EFT is given by
where U is a time evolution approaching the (future, say) singularity, and H , P are the
isometry generators corresponding to and translations in Schwarzschild coordinates.
(Of course, represents a spacelike direction near the singularity, and therefore H is really
a momentum here, despite the notation.) The U factor arises from the whisker region.
The exponential weight is a non-trivial consequence of our space time transformation,
where the timelike circles become standard spacelike circles. One can think of the
whiskerrelated factor, . . . U e 2 (H P), as setting up a useful out state inside the horizon of
the physical region.
If there are no sources (endpoints of correlators) in the vicinity of the singularity, the
time evolution U commutes with the isometry generators, H , P and hence cancels against
U . This cancellation, which also can be seen non-perturbatively in the CFT description, is
the deep reason behind the insensitivity of boundary correlators to the details of UV physics.
It matches the cancellations in Witten diagrams (before massaging by space time) in the
manner of (2.7). Such U U cancellation in the far future is reminiscent of what happens for
correlators in the in-in formalism [46, 47] (see [48] for a modern discussion and review).
Indeed, we will show using the space time transformation that local boundary correlators
traversing the singularity are equivalent to a generalization of in-in correlators involving
non-local operators, where all time evolution takes place after the past singularity and
before the future singularity.
Studying the singularity
Our ability to discover and check our proposal for describing scattering inside the BTZ
horizon rests on the existence of the protected set of local boundary correlators, which we
can prove in a simple way are insensitive to the singularity. However, the ultimate goal
is not to merely describe scattering inside the horizon far from the singularity, since such
scattering is approximately the same as scattering in a static spacetime. This regime is only
useful to vet our proposal, precisely because we know the answers already, dominated by
EFT. Rather the goal is to use our non-perturbative CFT proposal to describe scattering
close to the singularity where cosmological blueshifts take us out of the EFT domain, and
where even perturbative string theory may miss important features. This interesting kind of
sensitivity to the singularity is not outright absent from the protected set of correlators, but
it is suppressed by 1/blueshift. However, one can study processes with kinematics chosen
such that they would not proceed but for such cosmological blueshifts (that is, they would
not proceed for rS = ), in which case the leading effects are sensitive to the singularity.
Furthermore, more general (gauge-fixed EFT) bulk correlators are order one sensitive
to the singularity and UV physics, but not mathematically divergent. The same is also true
for local boundary correlators in the more extended AdSglobal/ realization of BTZ, as we
will discuss in [44].
Relation to the literature
Several earlier attacks have been made on more explicitly extending holography into the
black hole interior, some specific to BTZ, while others apply also to higher-dimensional
eternal black holes. The most direct approach has been to study the thermofield CFT
formulation carefully, and to identify those subtle, non-local features that might encode key
aspects of the black hole interior [19, 4951] (see [52] for higher-dimensional discussion).
Our work is certainly in the same spirit, but we claim our non-local CFT operators more
sharply and more knowably probe the interior. Another general direction is to try and
construct the CFT dual of interior field operators [5355], in part by using the gravitational
EFT equations of motion to evolve exterior field operators in infaller time into the interior.
This is necessarily restricted to situations in which the bulk metric fluctuates modestly,
whereas we propose a non-perturbative formulation. Yet another general approach is to
try to enter the horizon by a variety of analytic continuations of external (Lorentzian or
Euclidean) correlators [45, 5659]. Our work has this aspect to it, but it is governed and
understood from a physical perspective in which analytic continuation merely provides
an efficient means of calculation, rather than a first principle. The symmetry-quotient
structure of BTZ has led to attempts to construct a symmetry-quotient form of a dual
CFT [21]. Another BTZ-specific approach is to take advantage of being able to follow the
BTZ geometry beyond the singularity, where further AdS-like boundary regions exist.
One then tries to make sense of CFT on the various boundary regions and how they connect
together [56, 60]. Our work furthers these directions, of making sense of the quotient
structure from the CFT perspective, and using it to show how different boundary regions
are entangled. A number of variants of BTZ have also been constructed and studied [61, 62].
Organization of paper
We start from the symmetry quotient construction of BTZ from AdSPoincare, and try to make
sense of the idea of a quotient CFT dual. In section 3, we review the quotient construction
of BTZ geometry from AdSPoincare and how this extends the spacetime smoothly past the
singularity, although gravitational EFT diagrams ending at the singularity do diverge. In
section 4, we identify the boundary regions of the BTZ spacetime, outside the horizon and
inside the whiskers. We point out the central challenges for formulating a dual CFT on the
boundary of BTZ, related to the presence of lightlike and timelike closed curves. In section 5
we explore the BTZ singularity with the simplest examples, before beginning a more general
attempt to formulate a CFT dual. The relevant BTZ correlators, with end points inside
and outside the singularity and horizon, are obtained by the method of images applied
to AdSPoincare. We illustrate how naive divergences encountered as interaction vertices
approach the singularity in fact cancel to give mathematically well-defined correlators. In
section 6, in order to massage the CFT on the BTZ boundary into a non-perturbatively
welldefined form, we introduce the transformation switching time and space inside the horizon,
arriving in (6.11) at our central result, a generalization of the thermofield CFT formulation
allowing probes of physics inside the horizon. Eq. (6.11) is manifestly well-defined and
manifestly respects the symmetry construction of BTZ. In section 7, we recast (6.11) in
canonical thermofield form, resulting in (7.5), with probes inside the horizon appearing
as non-local probes of the thermofield-entangled CFTs. Many of our manipulations in
sections 6 and 7 are formally based on the CFT path integral. But for concrete confirmation
we must turn to the dual AdS diagrammatics.
In section 8 we study the Rindler AdS/CFT correspondence (rS = ) in detail, and
prove the above results in this limit in bulk EFT, allowing us to probe inside the Rindler
horizon by studying specific non-local correlators outside the horizon. We check that our
proposal reproduces the AdSPoincare correlators everywhere. In section 9, we finally check
that eq. (6.11) does indeed act as the dual of BTZ by showing that it gives the associated
local boundary correlators, including the whisker regions, and that these correlators are
finite and dominated by EFT (despite traversing the singularity). This follows from the
analogous Rindler proof in section 8 by applying the method of images in EFT. We explain
how these local boundary correlators are generally insensitive to the breakdown of EFT
near the singularity, allowing us to use EFT to check our CFT proposal is sharply sensitive
to scattering inside the horizon just as in the Rindler (rS = ) limit. In section 10,
we demonstrate that bulk correlators are sensitive to the singularity and UV physics
there, although still mathematically finite. We also show how to design special boundary
correlators where the near-singularity UV physics dominates, so that our CFT proposal is
needed to describe them. In section 11, we comment on our derivations and some aspects
of the physical picture that emerges from our work, and outline future directions.
BTZ as quotient of AdSPoincare
In higher-dimensional black holes, the Kruskal extension into the interior ends at a curvature
singularity. In the BTZ case however, uv = 1 in (1.5) does not represent a true curvature
singularity and the geometry can be smoothly extended beyond it. Such an extension is
most simply given by a quotient of the Poincare patch of AdS (AdSPoincare) [34, 63, 64],
(x, z) (erS x, erS z).
As straightforwardly checked, the Poincare coordinates are related to the Kruskal
coordinates by
z = e
r
The horizon, singularity and asymptotic AdS boundaries now reside at:
The true nature of the apparent black hole singularity becomes clearer. While the BTZ black
hole spacetime has locally AdS geometry and finite curvature everywhere, the singularity
surface consists of closed lightlike curves, given by x = t cos , z = t sin , parametrized by
. The region inside this surface consists of closed timelike curves, which we will call the
whisker region similarly to [37].
The presence of such curves does not in itself constitute a geometric singularity,2 but it
does pose a conceptual challenge for physical interpretation, and on general grounds implies
2In the BTZ realization as a quotient of AdSglobal the singularity also includes a breakdown of the
spacetime manifold (Hausdorff) structure itself. But the points at which this further complication takes
place are pushed off to infinity in our Poincare patch realization of BTZ. This breakdown is relevant to some
of the studies in the literature but not to the correlators discussed in this paper. We will more thoroughly
clarify this point in [44].
z = 0
x = 0
z2 x+x = 0.
the breakdown of quantum (effective) field theory in the vicinity of the closed lightlike
curves [40]. See [41] for a concise review, and [38, 39] for computations of stress-tensor
divergences at the BTZ singularity. Similar features have also been studied in string theory
(as reviewed in [42].) We illustrate the basic problem by looking at an EFT amplitude for a
scalar field in the BTZ background. Following [45] we focus on the scalar propagator from
a point on an AdS boundary, x, external to the black hole to a point inside the horizon
and near the singularity, (y, z). Because BTZ is a quotient of AdSPoincare, we can easily
work out this propagator by the method of images applied to the boundary-bulk propagator
of AdSPoincare [33, 45]:
where m2 = ( 2) and we have summed over images of the bulk point. Generically, the
image sum clearly converges, the summand behaving asymptotically as e|n|rS. The
exception is the singular surface z2 y+y = 0, where the summand becomes n-independent
for large n, and the series diverges. (We omit a discussion of the i prescription in the
propagator as it does not avoid the divergence as 0, although it will play an important
role later in the paper.) This feature is general for correlators with some points ending on
the surface z2 y+y = 0, the perturbative incarnation of divergent backreactions that
justifies this surface being called the singularity. A subtler question is whether one can
propagate or scatter through the singularity within gravitational EFT. If so, one can just
avoid probes (correlator endpoints) very near the singularity and trust EFT calculations
elsewhere. This question is particularly relevant for correlators ending on the full BTZ
boundary (including inside the whiskers) since these would define local operator correlators
of a possible CFT dual to BTZ. Before tackling this question, we first study the BTZ
boundary itself.
The extended BTZ boundary and challenges for the CFT dual
Since BTZ is at least locally AdS-like, it seems very natural to guess that BTZ quantum
gravity has a holographic dual given by a 1 + 1 dimensional CFT living on the BTZ
boundary. Within the Kruskal extension, this boundary, uv = 1, consists of the two
disjoint solutions with u > 0, v < 0 or u < 0, v > 0, corresponding to the two asymptotically
AdSglobal regions outside the horizon, as in higher-dimensional AdS black holes. This is
then consistent with the now-standard thermofield picture of two CFTs living on two copies
of the boundary of AdSglobal, namely two separate CFTs each living on a spatial circle
infinite time, but in an entangled state. However, this Kruskal boundary corresponds in
our AdSPoincare coordinates to z = 0, x+x < 0, whereas in the AdSPoincare realization the
boundary is straightforwardly all of z = 0. The regions z = 0, x+x > 0 are missed in the
Kruskal extension because they lie inside the singularity, while the Kruskal extension stops
there. The question then arises whether these inside-singularity boundary regions play an
important role in the CFT dual of BTZ (the view taken in [56, 60]), even for projecting
the part of BTZ outside the singularity but inside the horizon. We will show that there
are in fact two equivalent formulations of the CFT dual of BTZ, one in which the entire
boundary region is needed for the CFT, and a second one in terms of two entangled CFTs
on just the disjoint boundary regions outside the horizon.
BTZ boundary as disconnected cylinders
We begin by identifying the full boundary region of BTZ, BTZ, within the AdSPoincare
realization, regardless of where this takes us with respect to the singularity. Even the simple
identification of BTZ as z = 0 is subtle because of the quotienting. Naively this would yield
1 + 1 Minkowski spacetime with the identification x erS x. Such an identification does
not make straightforward sense because rescaling is not an isometry of Minkowski space.
The subtlety is that the boundary geometry is only determined from the bulk geometry up
to a Weyl transformation [10], which conformally invariant dynamics cannot distinguish.
Therefore more precisely,
ds2BTZ = f (x)dx+dx,
where f is a Weyl transform of 1 + 1 Minkowski spacetime, with the identifications
x erS x and hence f -periodicity f (x) = e2rS f (erS x).
Two choices of f will prove insightful. The first is
f = 1 . (4.2)
|x+x|
It is useful to break up the 1 + 1 Minkowski plane into the four regions,
so that the quotienting and Weyl transformation take the simple forms
We can simply restrict to the fundamental region rS rS. We see that BTZ is
then given by four disjoint spacetime cylinders,
ds2BTZ =
(+d+d, R, L spacelike circle infinite time
d+d, F, P infinite space timelike circle!
(a) The fundamental region in
mapped to the Minkowski plane
by the Weyl transformation (4.2).
(b) The fundamental region in
mapped to the Minkowski plane
by the Weyl transformation (4.7).
The cylinders in the Rindler wedges are just the boundaries of the AdSglobal asymptotics
outside the horizon described above. But the cylinders in the Milne wedges are the boundary
region inside the singularity.3 The Weyl transform maps the cylinders to the four shaded
regions of 1 + 1 Minkowski spacetime in figure 4a. In this way we can think of the shaded
region as a fundamental region for the quotienting procedure on the CFT-side.
We see that the four cylinders present two challenges for hosting a dual CFT. The
first is that they remain disjoint and therefore we need some sort of generalization of the
thermofield entanglement of two CFTs with which to connect them. For related early
work in this direction, see [56]. The second issue is that the Milne wedge cylinders have
circular time.
Connected view of BTZ
To guess how to move forward we use a different choice of Weyl transformation, which gives
us a different view of BTZ (the CFT being insensitive to such choices),
Using polar coordinates on the Minkowski plane,
with the usual identification + 2 and the BTZ quotient identification + 2rS,
we find (figure 4b):
ds2BTZ = cos 2(d2 d2) + 2 sin 2 dd
= Lorentzian Torus!
3These four disjoint boundary components are the AdSPoincare subset of the larger set of boundary
components arising in the further extension of BTZ as a quotient of AdSglobal.
Figure 5. The Lorentzian torus contains closed timelike (red), lightlike (yellow), and spacelike
(blue) curves. The inner and outer edges of the annulus are identified after Weyl transformation (4.7)
to make the Lorentzian torus of (4.9).
This geometry [65] is smooth and connected, but still contains alarming features
(figure 5). There are still timelike circles in the Milne wedges 4 < < 34 , 4 > > 43 ,
oriented in the -direction. (There are still only spacelike closed curves in the Rindler
wedges.) The joints, = 43 , 4 , 4 , 34 , while smoothly connecting the geometries of
the different wedges, are themselves light-like circles in . So it does not appear that a
CFT on this boundary region will allows us to evade the difficult problem of doing field
theory at lightlike circles presented by the BTZ singularity [40, 41]. (For discussion of a
very similar 1 + 1 context see [66]).) If we try to excise these lightlike circles we are left
with the disjointedness of the boundary and the CFT living on it.
Before making any interpretation, we will first try to simply define local operator
correlators of the CFT on the full BT Z, using the method of images applied to AdSPoincare
boundary correlators. However, we must check that these are even mathematically
welldefined in the face of the BTZ singularity. Therefore we first study simple examples and then
general features of how the singularity enters into correlators. We then show that boundary
correlators on all of BT Z are mathematically well-defined, although the singularity does
represent a breakdown of gravitational EFT.
Boundary correlators and the singularity
In this section we study the simplest examples which illustrate the implications of the
singularity for defining correlators within gravitational EFT, and for identifying them
with equivalent CFT correlators. For this purpose we will not need to study these BTZ
correlators in a UV-complete framework such as string theory, although we assume such a
framework exists. We will compute these BTZ correlators using the method of images. It is
convenient to define
so that the quotient identification (3.2) can be written
Approaching singularity from outside
We start by noting the full i structure of the bulk-boundary propagator of AdSPoincare,
which is important for what follows here:
KAdS =
This structure for Lorentzian K is most easily derived from the well-known Euclidean
K [10] by analytic continuation in time. The boundary point, x, z0 = 0 can be in any
of the boundary regions, L, R, F, P . The analogous BTZ propagator is given by summing
over images of the bulk point, enrS y, enrS z, as in (3.6). It is important that the i is also
thereby imaged. To study the near singularity region it is useful to follow [45] and switch to
AdS Schwarzschild coordinates, where the bulk point is at , , r, and the boundary point
is given by 0, 0, r0 = . We can zoom in on the region where the bulk point approaches
the singularity, r 0, and the image sum divergence for positive large n dominates:
KBTZ r0,
The approximation in the first line is to drop terms even more subdominant for large n > 0.
In the second line we noted that the n-dependent terms are subdominant for small r
for the first ln(/(er)) terms in n > 0, with the sum rapidly converging for larger n.
Therefore, the sum is given by the n-independent constant multiplied by ln(er), for
small r. Crucially, the i appears inside the logarithm by the first lines analyticity in
er + i .
Flawed attempt to scatter through singularity
Let us now explore the possibility that a dual CFT resides on BTZ, as identified in the
previous section, by trying to construct the leading in 1/NCFT planar contribution to a
3-point local operator correlator in terms of a tree level BTZ diagram:
where as usual, on the left-hand side the operators are defined operationally as limits of
bulk fields,
O = lim . (5.6)
z0
We choose two scalar primary operators to be on the R Rindler wedge, and the
remaining operator to be in the F Milne wedge, so that the diagram is forced to pass
through the singularity and we can test what difficulties it poses. In this section, we will seek
to understand if such a correlator is even mathematically well-defined, not yet addressing
its physical interpretation, given that one operator lies inside the singularity where there
are time-like closed curves. In subsection 7.2 we will give a physical interpretation of the F
endpoint as associated to a conceptually straightforward but non-local operator in a hot
(thermofield) CFT. For convenience, we have chosen the F scalar primary to be different
from the R operators with different scale dimension, 6= , so that there are two dual
scalar fields in BTZ. We consider a typical non-renormalizable interaction term in BTZ EFT,
rS/2
ln(r + i )
d (function of , ) r (r + i )2 0 . (5.8)
Ref. [45] earlier worked through a very similar calculation. Naively, this blocks us from
defining such correlators. However, this calculation is in error.
Approaching singularity from inside
Schwarzschild coordinates are useful for cleanly separating out the direction of approach to
the singularity from the direction which is being imaged, but they are restricted to only the
outside of the singularity, z2 y2 > 0. We should also include the asymptotic contributions
as the interaction vertex approaches the singularity from inside it, z2 y2 < 0. This requires
new Schwarzschild-like coordinates for z2 y2 < 0:
ds2 = (r2 + 1)d 2 r2dr+2 1 + r2d2.
as long as the diagram (propagator) remains analytic along the neighborhood of the path
traced out thereby in the complex x, z planes, ultimately arriving after k iterations to
x x and z iz.
This is indeed the case, as we now check. The bulk AdSPoincare propagator has the
form [74, 75],
where F (2) F ( 2 , 2 + 12 ; ; 2) is a hypergeometric function which is analytic in the
complex 2-plane with a cut along (1, ), and where
Because of the branch cuts in F and we must be very careful in any analytic continutations
we perform. Our first step will be to simply rotate all z coordinates in the diagrams of (8.18),
It is straightforward to check that never passes through a branch cut of GAdS in such a
rotation.
Let us interpret this move. If z corresponds to a point on the T -hypersurface, then
this rotation is just part of the action of e 2 (SK) acting on |N i, as discussed above. The
e 2 (SK) is also supposed to rotate the associated x, but since we are taking T very large,
and well to the future/past of our sources, x 0 along this hypersurface. Therefore the
action of e 2 (SK) on it is trivial. If instead, z corresponds to an interaction vertex, then
this move corresponds to a (passive) contour rotation of the integral over the interaction
vertex location. The only other possibility is that z corresponds to a source point, where
J 6= 0. For a source localized to the AdS boundary, necessarily z = 0, which is insensitive to
the rotation. For a bulk source which is analytic enough in z, the above move would again
correspond to (passively) rotating the contour of the z-integral over the source region. We
will discuss subtleties of boundary and bulk source terms further in subsections 8.5 and 8.6,
respectively, but proceed with allowing rotation of source points for these broad reasons.
After completing the above rotation of all z coordinates in the diagrams of (8.18), at
= it is straightforward to check that we end up with
From this, and the fact that the hypergeometric function in terms of which G is given
satisfies F (2) = F (2), we obtain the simple but non-trivial identity,
G x, e i(2 ) z; y, e i(2 ) z0 = G(x, z; y, z0).
The contour rotations of interaction vertices for real to (nearly) imaginary z results in the
change of integration measure,
resulting in the replacement in diagrams
2 Z dz
2 Z dz
(i couplings) (+i couplings).
We see that both propagators and interactions are thereby complex-conjugated, and the
sign of every x is flipped in the diagrams corresponding to (8.18). This all happened as
a consequence of a single active move, namely to act with e 2 (SK) on the points ending
on the T -hypersurface. The complex conjugation simply undoes the conjugation already
appearing in (8.18). For interaction vertices x x is clearly irrelevant since it is
integrated, and for points ending on the T -hypersurface we are insensitive to x x
because x 0 there. Therefore, x x is only significant for source points. This now
corrects the naive wrong, that we started with F -wedge sources for in the R-wedge, as
noted below (8.15). The action of e 2 (SK) has performed this correction.
We are now poised to recover all AdSPoincare correlators from our thermofield formula,
but must carefully consider boundary versus bulk source options.
Testing boundary localized correlators (in all regions)
we are not integrating z. Therefore rotating such z as we prescribe in the previous subsection
will not be a passive move, but will result in an extra factor of 1/i from the above limit.
This is easily corrected by multiplying the correlator from the trace formula by i for each
external boundary point in the F (P ) region. Then, for boundary sources, the diagrammatic
analysis of the previous subsection proves that (8.17) is
hN |eiKT e 2 (SK) T UF |i = h|
As discussed below (8.25), the x x applies to all source points in UF , correcting
the naive wrong of starting with F -wedge sources for in the R-wedge. A completely
analogous analysis can be made for the P wedge. Eq. (8.13) thereby takes the form,
ZRindler Thermofield =
hN |eiKT T ULUReiKT |M i
hM |eiKT
where we used the orthonormality of |N i to get to the second equality, and the fact that all
future and past operators lie to the future and past of the L, R wedges respectively, and that
L wedge operators commute with those of the R wedge, to get to the third equality. Again,
the {xF, xP} {xF, xP} applies only to source points in UF and UP , correcting the
naive wrong. We have thereby demonstrated that our trace formula and its thermofield
equivalent correctly reproduce arbitrary (local) CFT correlators in Minkowski space as
captured by the dual AdS EFT.
The i factors needed to achieve the above agreement may seem unusual, but they are
just what one should expect of a conformal transformation law of a scalar primary O, given
our improper conformal transformation,
O0 =
x0 = t, t0 = x or x0+ = x+, x0 = x. Equivalently, in the F wedge, local operators O in
the trace formula are reinterpreted as
e 2 (KS)O(x)e 2 (SK) = e i2 O(x),
inside T -ordered matrix elements. Therefore there is a perfect match between our trace
formula and the Minkowski/Poincare formulation once these transformation factors are
included.
In detail, we see that the F, P source terms in the trace formula must have extra i
factors in order to yield a desired set of AdSPoincare source terms. If we think of source terms
as perturbations of the CFT Hamiltonian, then hermiticity of such perturbations implies
that J is real for hermitian O. Clearly, to get such sources for the AdSPoincare correlators,
we must start with complex sources (i real) in the trace formula, corresponding to
nonhermitian CFT perturbations there. This appears to be an essential part of our construction
following from the improper nature of the conformal transformation switching x and t. We
will see a generalization of this feature for bulk sources.
Testing general bulk correlators
Finally, consider bulk source terms in F . As mentioned in subsection 8.4, this case is easiest
if we have a bulk source which is analytic in z. Suppose our goal is to end up with a bulk
correlator with a F region source
J = Z0 dzz e a12 (log zlog z)2 (t = t, x = x, z)
J (t, x, z) = (t t)(x x)z2e a12 (log zlog z)2
This is a nice Gaussian function of proper distance in the z direction, with size set by a,
which can be as small as desired. Note that this source term is analytic in z throughout
the set of rotated values in (8.21), and falls off rapidly as |z| 0, . To obtain such a
source for our AdSPoincare correlator, we have seen in subsection 8.4 that we must begin in
the trace formula with a source which analytically continues to the target source above, as
z iz. That is, in the trace formula we must begin with
J = Z0 dzz e a12 (log zlog zi 2 )2 (t = x, x = t, z)
J (t, x, z) = (t x)(x t)z2e a12 (log zlog zi 2 )2
As discussed earlier, the trading of x and t will be fixed by the action of e 2 (SK). The
analytic z integrand clearly becomes the target source integrand upon performing the
z iz move of (8.21).
Again, what is unusual about such a source term for a real bulk field is that it is not
real, and therefore corresponds to a non-hermitian perturbation of a (diffeomorphism
gaugefixed) bulk Hamiltonian. Of course, one can break up such complex sources into real and
imaginary parts, so that we reproduce our target AdSPoincare correlators/sources by taking
straightforward complex linear combintations of the corresponding trace formula correlators.
With this slightly non-trivial matching of source terms, the results of subsection 8.4 again
translate into the trace formula reproducing the AdSPoincare correlators (integrated against
the target sources).
The non-trivial matching of sources is to be expected once we take into account that
the trace formula reinterprets x t in the CFT in the F region (and similarly for P ), the
result of e 2 (KS)O(x)e 2 (SK) for any operator O whether local or non-local. When a
bulk field operator (in some diffeomorphism gauge-fixed formulation of quantum gravity),
(x, z), corresponds to some kind of non-local CFT operator by AdS/CFT duality, it
should be reinterpreted in the trace formula as
e 2 (KS)(x, z)e 2 (SK) = (x, iz),
if it lies in F , inside T -ordered matrix elements. Of course the bulk field for imaginary z on
the right-hand side is not a priori well-defined, so this equation should be thought of as a
short-hand for our main result: for analytic sources the AdSPoincare sources match the trace
formula sources via continuation z iz for F /P regions.
Finite rs: BTZ/CFT
Finiteness of BTZ EFT correlators
We consider bulk or boundary correlators of the BTZ black hole, with sources anywhere in
the extended spacetime (including inside the horizon, or even beyond the singularity in the
whiskers), as long as bulk sources are analytic in z in the manner discussed in subsection 8.6.
In the gravitational EFT these BTZ correlators are obtained by the method of images
applied to AdSPoincare, in particular the (scalar) propagator in BTZ has the form,
GBT Z (x, z; y, z0) =
where, as in section 5, we define for convenience
n=
The second line of (9.1) follows from (8.19) and (8.20), where
A central question is the mathematical finiteness of such EFT correlators, given that
the associated Feynman diagrams generally traverse the singularity. This can be understood
by looking at the large image-number contributions in the above sum, where
n n n(iz2 x2i+zxz0) + O(1) ,
implies that for generic points the summand nF (0) for large n and hence the sum
converges rapidly. However, at the singular surface, z2 x+x = 0, if we neglect the i ,
we see that n and hence the summand, become n-independent for large n, and the sum
diverges. This is the diagrammatic root of the singularity. Once we take into account the i
term we see that we always get a convergent sum again, but for diagrams to remain finite
after the ultimate 0 requires major cancellations before that limit is taken. We studied
the simplest examples of this situation and such cancellations in section 5, but in general
correlators the requisite cancellations are not immediately apparent. Nevertheless they do
take place, as we now show in a simple and general way.
Let us again perform the complex rotation of all z coordinates as we did in (8.21), but
now stopping at an intermediate value of = /2,
As discussed in section 8, this simply represents a passive deformation of z-integration
contours in the complex plane for interaction vertices and bulk endpoints (with analytic
sources as in subsection 8.6), and multiplication by some phases for boundary endpoints.
Therefore this move does not affect the finiteness of the correlator. But now we see that
for all points in BTZ, we have
Figure 11. Relationship between bulk tree level BTZ diagrams and the corresponding diagrams on
the AdSPoincare covering space. The dark gray lines are to be interpreted as propagators inside the
gray solids (although they may end on the surface).
so that the propagator summand nF (0) always for large n, the sum converges, and
the correlator is indeed finite (even as 0). It is crucial to note that this finiteness
required integrating over all z > 0 in the first place, so that inside the horizon we are
integrating both inside and outside the singularity. Therefore finiteness required inclusion
of the whisker regions.
The relationship between BTZ and the covering spacetime AdSPoincare diagrammatics
is most straightforwardly seen in the (leading) tree-level diagrams of EFT, as illustrated in
figure 11. We draw the BTZ spacetime as filling in the Lorentzian torus, to topologically
make a solid torus with the Lorentzian torus surface as its boundary. In order to view BTZ
like this we have switched the roles of the two circles of the Lorentzian torus with respect
to figure 4. Specifically, we generalize (4.8) to the bulk,
We compare diagrams in the solid torus with diagrams in AdSPoincare, which we view in the
above coordinates as a solid Lorentzian cylinder by first removing the origin. Its boundary,
the surface of that cylinder, is interpreted as 1 + 1 Minkowski spacetime with the origin
removed in , coordinate space. In this representation, the solid torus is simply the
quotient of the solid cylinder by a discrete translation, periodizing the direction along
the cylinders length. Figures 11b and 11d show tree diagrams on AdSPoincare (as the solid
cylinder) where the endpoints of both diagrams are (examples of) images of the same set of
endpoints for a BTZ (solid torus) correlator. Wrapping the AdS diagrams onto BTZ in
figures 11a and 11c, the two AdSPoincare diagrams appear as different contributions to the
same BTZ correlator, but with different image terms for one of the propagators. In this way,
by adding up all connected tree AdSPoincare diagrams with end points being images of the
desired BTZ correlator, we get the tree-level BTZ diagram, where every BTZ propagator is
a sum over AdS image propagators.
Local boundary correlators: EFT dominance and scattering behind the
horizon
While EFT correlators are finite in BTZ, as described above, this in itself does not prove
that these finite correlators dominate the true correlators, which may also include the
contributions of heavy states of the UV complete quantum gravity. It is also not immediately
obvious that these BTZ correlators sharply probe scattering processes inside the horizon
in the same way that AdSPoincare correlators probe scattering behind the Rindler horizon.
However, both these properties are indeed true of the protected set of local boundary
correlators of BTZ realized as a quotient of AdSPoincare. (Bulk correlators contain extra
UV sensitivity, as do the more general boundary correlators in the BTZ realization as a
quotient of AdSglobal [44].)
We first demonstrate that for erS 1, local boundary EFT correlators are
dominated by n = 0 in the sum over images in each propagator, (9.1). This follows after
rotating z by = /2 in the complex plane, (9.5), so that for large and x, z, y, z0 O(1)
in propagators, all other terms are O(|n|). The scaling, x, z, y, z0 O(1) in
follows, even though these arguments are being integrated, if the boundary endpoints xi
(which determine the region of convergence of the integrals) are chosen O(1). That is, after
rotation of z, it is as if there were no singularity, just a very large but compact direction,
and the resulting diagrams are dominated by the equivalent un-imaged diagrams in the
covering AdSPoincare spacetime. In particular, since these un-imaged diagrams describe
scattering behind the Rindler horizon, the BTZ correlators must describe scattering behind
the quotient of the Rindler horizon, namely the black hole horizon.
Because we are limiting ourselves to the Poincare patch of AdS and its quotient, we
are restricted in how sharp scattering processes can be when initiated and detected from
the boundary. The reason is that we have to send and receive scattering waves from the
boundary at z = 0, naively suggesting a violation of z-momentum conservation. Indeed,
z-translation invariance is broken by warping but this does not allow us to scatter waves with
z-wavelengths much smaller than the AdS radius of curvature using boundary correlators.
On the other hand, there is no similar obstruction to how small the x-wavelength can be.
Wavepackets with z-wavelengths of order RAdS and much smaller x-wavelengths can be
aimed so that scattering definitely only takes place inside the horizon, and predominantly
away from the singularity. They thereby give us access to reasonably sharp probes of
inside-horizon scattering, but obviously not the most general scattering processes. In
short, the sharpness of BTZ boundary correlators is the same as for AdSPoincare boundary
correlators.
Since we are dominated by the un-imaged AdSPoincare correlators, with O(e|n|rS)
corrections to ensure BTZ compactness (in ), it follows that EFT dominates the boundary
correlators as it does in AdSPoincare. Even if we included a very heavy particle into the
Feynman rules, it can be integrated out in the leading n = 0 contribution as in AdS,
inducing only contact effective interactions among the light EFT states. We will see that
this UV-insensitivity is not the case for the subleading O(e|n|rS) effects in section 10,
and that the effects of large cosmological blueshifts near the singularity are indeed present.
It may appear that bulk correlators are similarly protected by the above reasoning,
but it is important to understand why this is not the case. The subtlety is that the above
analysis required first performing the complex rotation of (9.5). As we have seen, this only
changes Witten diagram contributions to boundary correlators by complex phase factors,
so that estimates for the magnitudes of different contributions apply straightforwardly to
the original correlator before rotation. However, this is not the case for bulk correlators,
where bulk sources have to be analytically continued to accomplish (9.5), as discussed
in subsection 8.6. In general, such analytic continuations will completely change the
magnitudes of different contributions. Therefore estimates performed after (9.6) do not
apply to the original BTZ correlators before (9.6). Indeed we will give an example of bulk
correlator UV sensitivity in section 10.
If the circle were always very large there would be no surprise that the correlators
approximate those of non-compact , namely AdSPoincare. But it is at first surprising
here that the n 6= 0 corrections are small even for Witten diagrams passing through the
singularity, where the physical size of the circle is going to zero, as seen in the Schwarzchild
metric of (1.4). We will see the deeper reason for this in subsection 9.5.
Method of images applied to Rindler AdS/CFT
We now use the method of images to go to the finite rS (compact direction) analog
of (8.13), relating local EFT correlators anywhere in BTZ to (non-local) EFT correlators
in two copies of the outside-horizon BTZ with thermofield entanglement:
ZBTZ = HHh| h1 (T UF )i eP eP
h(T UL) (T UR)i eP eP h(T UP ) 1i |iHH.
The left-hand side is the generating functional of the bulk or boundary correlators of the
BTZ black hole discussed above, with any bulk sources being analytic in z. The right-hand
side is written in terms of the thermofield state formed by two copies of the outside-horizon
(r > 1) portion of the Schwarzschild view of the BTZ black hole. (Of course these two
copies can then be thought of as the outside-horizon portions of a single extended BTZ
black hole.) This outside-horizon geometry is just the quotient of the AdSRindler wedge of
AdSPoincare. The time and space translation generators on the right-hand side are with
respect to the , directions of the Schwarzschild coordinates for the BTZ black hole, and
the fields in all source terms on the right-hand side live only outside the horizon.
The derivation of (9.8) from (8.13) is more transparent when the right-hand side is
written in trace form,
where the trace is over the Hilbert space on one copy of the outside-horizon region. This
equation is just the quotient of the analogous non-compact statement, where the left-hand
side is the generating functional for AdSPoincare correlators and the right-hand side is a trace
over the Hilbert space on AdSRindler. On both sides, the compact result follows by imaging
the relevant type of propagator and keeping coordinates within a fundamental region. As
pointed in the discussion below (6.12), the exponential weights in (9.9) are a net suppression
of high energy excitations of the Schwarzchild spacetime (outside the horizon), and therefore
the right-hand sides of (9.8) and (9.9) are mathematically well-defined, matching the good
behavior we have found for the left-hand side.
As discussed below (7.5), one can think of local correlators ending inside the horizon
(including whiskers) on the left-hand side of (9.8) as being equal to correlators outside the
horizon for non-local operators of the form eP OlocaleP on the right-hand side. So far
we have established that local boundary correlators in BTZ are EFT-dominated and finite,
but we still have not given a physical interpretation of such correlators when they end in
the whiskers, problematic due to the time-like closed curves. However, for local boundary
correlators, the right-hand side of (9.8) gives such a simple interpretation. Defining states,
|P ioutside eP eP h(T UP ) 1i |ioutside
|F ioutside eP eP h(T UP ) 1i |ioutside,
Eq. (9.8) can be re-written
That is, the correlators including possible endpoints in the whisker boundaries are equal
to correlators with endpoints only on the boundaries outside the horizon, but with the
thermofield state being replaced by the modified |P,F i states. Given that we have
established that such correlators are dominated by the non-compact AdSPoincare EFT
correlators (image terms being subdominant), we can readily interpret these new states.
In non-compact correlators, endpoints in the F (say) boundary just act to detect the
results of earlier scattering inside the Rindler horizon, or evolving backwards, they set up
out states, |F i which sharply probe the results of the scattering process. The same must
therefore be true after quotienting to BTZ, where the F boundary is the whisker boundary.
In summary, the whisker regions can be thought of as an auxiliary spacetime in which the
local boundary correlator endpoints encode non-local operators that sculpt the thermofield
state into a variety of in and out states that probe the results of scattering inside the
horizon, very much as do F /P boundary correlators in AdSPoincare. Furthermore, the local
boundary correlators of BTZ are diffeomorphism invariants of quantum gravity.
Connecting to CFT dual on BTZ
It remains to connect (9.8) to the CFT thermofield form of (7.5), (7.10), or equivalently the
CFT on the Lorentzian torus BT Z. The diagrammatic expansion in the bulk theory
is dual to a large-NCFT expansion in a CFT gauge theory. At infinite rS, tree diagrams
such as figure 11b capture the same physics as the planar diagrams of figure 12b in the
dual CFT, by standard AdS/CFT duality. Just as figure 11b maps to contributions to
BTZ correlators for specific fixed images in figure 11a, figure 12b maps to planar diagrams
of the CFT on the Lorentzian torus in figure 12a, as discussed in more detail for the
Figure 12. Relationship between planar CFT diagrams in double-line notation (reviewed in [12])
on the Lorentzian torus and its covering space, the Lorentzian cylinder. These CFT gluon lines
are to be interpreted as propagating on the boundary surfaces of the gray solids in figure 11. The
black dots represent local CFT operators.
example of subsection 5.5. Equivalently, we have seen that we can use the right-hand
side of (9.8) for BTZ tree amplitudes, and these are then identified with the careful
construction of (6.10), (6.11) for the CFT on the Lorentzian torus at planar order. That
is, the method of images straightforwardly identifies the bulk tree amplitudes to planar
CFT amplitudes, either directly on ()BTZ or in equivalent thermofield form. The value
in the CFT construction of (6.10), (6.11) however is that it includes a UV-complete and
non-perturbative (in 1/NCFT ) description of the approach to the singularity, even when
bulk EFT eventually completely breaks down.
Naively, at the same planar order in the CFT there are also diagrams which wrap
around the torus in the direction, such as figure 13a, which do not descend from cylinder
(Minkowski) planar diagrams by the method of images, and yet are of the same order
in NCFT . These diagrams necessarily break up a minimal color singlet combination of
gluons and send some of them to an operator and the remainder to its image (figure 13b).
But in the full gauge-invariant path integral on the torus such diagrams are constrained to
vanish. This is a familiar fact if we think of the direction as time (now that we are
acclimatized to choosing the time direction for our convenience): the non-abelian Gauss
Law constraint says that only gauge-invariant states propagating around the direction are
physical, whereas any part of of a minimal color singlet cannot be gauge-invariant. Closely
analogous to this, in equilibrium thermal gauge theory it is the Gauss Law constraint
that enforces that only gauge invariant states can circle around compact imaginary time
(equivalently, the thermal trace is only over gauge-invariant states).
At nonplanar order in the CFT, there are subleading diagrams that can be identified
with the loop-level bulk diagrams that unitarize the tree-level contributions. But there
are also new CFT contributions not of this form, namely creation and destruction of
finite-energy Wilson-loop states winding around the compact direction. These have
no analog in the non-compact case. In BTZ, these are dual to quantum gravity states,
Figure 13. Naively, there are diagrams at leading order in NCFT, such as (a), but which unwrap
to diagrams in Minkowski space, such as (b), which violate gauge invariance (for example, gauge
non-singlets are created by different images of the same operator). Such contributions vanish by
gauge invariance.
generically finite-energy strings, that wind around the bulk circle, but which have no
analog in non-compact AdSPoincare. The effects of such extended objects cannot be captured
by the simple diagrammatic method of images we have followed for BTZ. If the extended
objects have tension then the winding states will ordinarily be extremely heavy for large
rS, and thereby give exponentially suppressed virtual contributions to correlators between
well-separated source points. But approaching the singularity, the physical -circumference
approaches zero, as seen in the Schwarzchild metric, (1.4), and the winding states can
become light. They are then part of the normally-UV physics which becomes important
near the singularity. See [43] for discussion within string theory. We expect this physics to
be contained in our CFT proposal for the non-perturbative BTZ dual, but not part of the
EFT checks we have performed in the regime where we argued EFT should dominate.
Finally, beyond any order in 1/NCFT , the CFT correlators will have effects, not
matching bulk EFT or even perturbative string theory. They may well play an important
role near the singularity.
Deeper reason for insensitivity to singularity
Our diagrammatic derivations have non-trivially confirmed our formal CFT expectations
for local correlators set forth in (7.5). But this does not explain why EFT is well-behaved
despite the singularity, why technically there was a way to deform the contour for interaction
vertex integrals so as to avoid the perturbative face of the singularity in image sums, and
for boundary correlators why these image sums converge so rapidly. We might also worry
that EFT misses important UV physics near the singularity, such as heavy particles or
the winding states mentioned above. In general, we therefore want to understand whether
to trust EFT at all for boundary correlators, especially when some endpoints are in the
whiskers.
We begin with non-compact AdSPoincare where we understand the diagrammatic
expansion. We have formally motivated and then diagrammatically derived the Rindler
AdS/CFT result of (8.13) and more compactly, (7.5). The typical local boundary correlator
of AdSPoincare is thereby re-expressed on the right-hand sides using (7.6),
h|{1 T [(eP O1F eP ) . . . (eP OnFF eP )]}
where the O are local Heisenberg operators of the CFT or local boundary operators of AdS.
The operators of the form eP OeP are then non-local, but only in the spatial sense.
Time evolution, implicit in the Heisenberg operators, ranges between the earliest and latest
times that appear in any of the O operators above, early, late, say. It is important to note
that this time evolution does not go all the way from = to = +. This is no
surprise because we have a (generalized) in-in formalism [46, 47] (see [48] for a modern
discussion and review) in the thermofield form for correlators. We represent this situation
in the Penrose diagram of figure 14, where the symmetry direction is omitted, but is now
non-compact < < . The spacelike hypersurfaces are pinned on the boundary by the
boundary time evolution, but their form in the bulk is otherwise arbitrary by diffeomorphism
invariance. What is not immediately obvious from the figure, but straightforwardly verified
by the AdSPoincare metric, is that all such hypersurfaces pinned to the boundary outside
the (Rindler) horizon cannot go beyond the jagged lines at any point. This is the only
significance of the jagged lines in figure 14 since there is of course no singularity in AdS.
We have depicted the simplest choice of such hypersurfaces.
The central point when we move to the compact BTZ case for such correlators, as
in (9.8), is that figure 14 still holds, but now with the omitted direction of course being
compact, and the jagged lines depicting the location of the singularity. What we see is
that in deriving (9.8) from (8.13) we are only trusting the diagrammatic expansion and
the method of images to compactify away from the singularity. As long as early/late are
not too early or late, the physical circumference of the circle can be taken to be large
throughout the time evolution and it is not surprising if our correlators are dominated by
the non-compact limit and insensitive to the UV physics of the singularity. In particular,
winding states will be very massive throughout this evolution.
Sensing near-singularity physics
A good part of this paper has been concerned with the validity and use of bulk EFT
and the diagrammatic expansion in order to capture scattering processes behind the BTZ
horizon. This has allowed us to test our proposed non-perturbative CFT formulation under
conditions where we already know what to expect. However, the real importance of such
a CFT formulation is that it allows us to study processes close to the singularity, where
large cosmological blue-shifts make the physics very UV sensitive and EFT breaks down.
In this section, we want to demonstrate that the UV-sensitive physics near the singularity
is certainly present in the BTZ quantum gravity and that whisker correlators and their
CFT duals can detect this. To do this, we will show under what circumstances we become
sensitive to heavy states beyond EFT, and yet without such sensitivity invalidating our
derivations.
We know that we see divergences if correlator endpoints are right on the singularity, as
simply illustrated just by (3.6). But EFT should come with some effective cutoff length,
below which we do not ask questions. If we simply move correlator endpoints more than
the cutoff length away from the singularity they are finite and the cosmological blueshifts
are more modest. But mathematical finiteness is not necessarily the same as insensitivity
to heavy states. We begin by demonstrating that even at distances/times of order RAdS
away from the singularity, correlators are sensitive to the UV heavy states outside BTZ
EFT. To do this we move the point in the F wedge of our section 5 example correlator from
the boundary to the interior and consider
d2ydzg GBTZ(xF , z0; y, z)gMN M K(xR1 ; y, z)N K(xR2 ; y, z).
We assume from now on that 1 and corresponds (via m 2 = ( 2)) to some heavy
particle of BTZ quantum gravity that is more massive than the cutoff of BTZ EFT (string
excitations). We are going to show that we are sensitive to such states at order RAdS
separations from the singularity.
Choose xF , z0 to have timelike geodesic to some points on the singularity, with proper
times to these points < RAdS( 1), but much larger than the cutoff length. For example,
xF = (z0 ), < 1 and near-singularity points (y z0, z z0) are related in this way.
For such small separations, the bulk propagator can be approximated by its 2 + 1 local
Minkowski equivalent,
GAdS q
(xF y)2 (z z0)2 i (x0F y0)2
where z0 is the approximately constant redshift of the inertial 2 + 1 Minkowski patch. GBTZ
is of course obtained by images of (y, z) from GAdS. Combining this sum with integration
of interaction points over the fundamental region to get integration over all AdSPoincare,
d2ydzg q
(xF y)2 (z z0)2 i (x0F y0)2
gMN M K1N K2 + .
ei 22rr0 cosh(0)+(r2+r022) cosh(0)
Z r0 drdd r ei (r0r)2(0)2
r0 (r + i )2 p(r0 r)2 ( 0)2
with ( 0), ( 0), r, r0 all small, but only r 0. As r 0, timelike separation to
(xF , z) requires r02 > ( 0)2, so
The behavior at r0 is smooth so we are basically computing the Fourier transform of r+1i .
As 0, we have unsuppressed Fourier components and there is no suppression for large
. In other words the particles sent in from the R wedge are able to produce cutoff scale
heavy particles that can propagate far away from singularity. Therefore these heavy states
cannot simply be integrated out by r0, even though lcutoff 1 r0 < 1. So EFT cannot
be trusted at r0.
We can contrast this situation with with the analogous AdSPoincare correlators (not BTZ):
d2ydzg GAdSgAMdNS M K1AdSN K2AdS.
Without any infinite image sums, the Ks are smoothly varying on RAdS 1 length scales,
except on light cones from x1,2. We take (xF, z0) to be away from these lightcones. For
1 not to be the exponent of a suppression, (y, z) must be timelike separated with
separation < RAdS. Therefore in looking for unsuppressed contributions, (y, z) can also be
taken away from x1,2 lightcones. But then G rapidly oscillates on 1 lengths, so its integral
with the smooth KK is highly suppressed. This is the standard reason for why we can
integrate out heavy in long-wavelength processes. In AdSPoincare we do not see the kind
of breakdown of gravitational EFT that we see in BTZ.
It is important to note that the BTZ sensitivity to heavy particles, just illustrated, takes
place in a correlator with one bulk endpoint. Thus it is not in contradiction with our general
observation that the purely local boundary correlators are dominated by EFT. But it is
the local boundary correlators that are most straightforwardly matched non-perturbatively
to CFT correlators and ideally we want to use just these to detect the UV physics near
the singularity. Fortunately, while we have shown that EFT dominates local boundary
correlators, and even more strongly that the non-compact limit (AdSPoincare) dominates,
this does not preclude the UV physics from residing in the small corrections to these leading
approximations. The key then is to look at boundary correlators that vanish at the leading
approximation, so that the small UV-sensitive effects dominate.
In the process we have considered that creates a heavy particle near the singularity
using the large cosmological blueshift there, and propagates it into the whisker, the obvious
way to get a purely local boundary correlator is to attach two light particle lines to the
bulk point in figure 15a and then connect these to the whisker boundary, as in figure 15b.
Using the ability to choose the boundary sources for the four boundary points, we can insist
that the incoming beams are softer than the threshold for heavy particle production unless
one takes into account the cosmological blueshifts, that is unless one looks at large image
numbers in the propagators and not just n = 0 in the notation of (9.1). Similarly, we can
choose sources for the boundary whisker points to be looking for hard particles coming
from a point away from the singularity. In this way, we have chosen the boundary correlator
to vanish at the usual leading approximation of the non-compact limit, but clearly the
full correlator captures the heavy particle production near the singularity and its distant
propagation. In fact it might seem that this UV sensitive BTZ boundary correlator is order
one, in violation of our general result. But it is easy to see the source of suppression: if it
were not for the warp factor we would expect z-momentum to be conserved, in which case
the heavy particle produced from light particle beams originating at z = 0 would not decay
into light particle beams which return to z = 0. Instead we would expect the final light
particle beams to escape to large z and not contribute to this purely boundary correlator.
The warp factor can indeed violate z-momentum conservation, but it is a very mild effect
for hard incoming beams. This is the source of suppression of the boundary correlator
that is in keeping with our general result. Therefore, the four-point correlator depicted
in figure 15b is small, but the UV-sensitivity dominates this small correlator. Using (7.5)
and (9.8), we can write this correlator as a non-perturbatively well-defined CFT thermofield
correlator. Of course there might be other UV physics which is harder to model, which
would be picked up in similar fashion by our non-perturbative formulation. This is the
central payoff of our work.
Comments and conclusions
We have made a precise proposal for the non-perturbative CFT dual of quantum gravity
and matter on a BTZ black hole, in terms of 1 + 1 Minkowski CFTs with weakly-coupled,
low-curvature AdSPoincare duals, and provided several non-trivial checks. It extends the
nowstandard duality by making sense of a CFT living on the full BTZ boundary realized as a
quotient of AdSPoincare, which includes whisker regions beyond the singularity containing
timelike closed curves. We did this by observing that there are well-defined non-local
generalizations, eP , of the familiar Boltzmann weight, eH , which effectively switch
Figure 15. Sensitivity to the singularity. The cone marks the location of the singularity and the
dashed line represents a heavy particle. The lower black lines represent two incoming particles that
are initially subthreshold. The heavy particle can be produced due to blueshifting as the singularity
is approached. In (b), the heavy particle subsequently decays and its decay products are received at
the boundary.
the roles of space and time inside the horizon, and turn the timelike circles into familiar
spacelike circles. We then gave an equivalent thermofield construction of our CFT dual in
which non-local correlators in the entangled CFTs are responsible for capturing the results
of scattering inside the horizon, giving a concrete realization of complementarity.
We chose to realize BTZ as a quotient of AdSPoincare, rather than of AdSglobal, based
on its greater technical simplicity, and because the set of local boundary correlators in
this smaller spacetime are protected, in the sense of being dominated by gravitational
effective field theory even when the contributing Witten diagrams traverse the singularity.
This construction gave us the minimal extension of BTZ beyond the singularity to make
contact with boundary components within and to explore the role they play, even in just
ensuring the mathematical finiteness of bulk amplitudes. But both AdSPoincare, and the
portion of the extended BTZ spacetime it covers, are geodesically incomplete. Our CFT
proposal projects this geodesically incomplete portion of BTZ in an analogous manner to
the way in which CFT on Minkowski spacetime projects quantum gravity on geodesically
incomplete AdSPoincare. Our CFT dual of BTZ lives on the Lorentzian torus, which is also
incomplete because of geodesics that can escape by passing close to the lightlike circles.
But in our careful construction we are cutting out thin wedges around the lightlike circles
so this does not arise. Alternatively phrased, in our final construction we only use CFT on
spacetime pieces of the cylindrical form circle time. We will address the maximally
extended BTZ spacetime arising from the quotient of AdSglobal in future work.
While analytic continuation played a role in this paper, we believe it was a matter of
calculational efficiency, rather than as a conceptual tool. For example, in subsection 5.4
studying scattering through the singularity, we arrived at the same conclusion by direct
computation of BTZ diagrams and by rotating the interaction integral contour of the
z-coordinate. In section 8, we used analytic continuations as the simplest way of computing
the non-local consequence of the eP generalized Boltzmann weights. In principle one
could directly do the integral over such weights without any continuations but it would be
technically much harder. We have checked that the direct computation in free CFT gives
the same result as analytic continuation.
We believe our approach should be closely generalizable to quotients of
higherdimensional AdS spacetimes [7678]. These yield interesting black objects with horizons
and singularities. Of course it would be a greater technical feat to obtain the dual of
higher-dimensional black holes or higher-dimensional cosmologies, without the advantage of
a quotient construction from AdS, and with even worse (looking) singularities. It remains
of great interest to understand the dual of evaporating black holes. We hope that the Ising
model of black holes, BTZ, shares enough in common with other systems with horizons
and singularities to provide hints on how to proceed.
In the paper, we have viewed the whisker regions, in particular their boundary, as an
auxiliary spacetime grafted onto the physical spacetime which is useful in defining states
on the physical region, much as Euclidean spacetime grafts are useful in defining
HartleHawking states on physical spacetime. However, since the whiskers do have Lorentzian
signature, it is intriguing to also see if they can be accorded any more direct physical reality.
Once the whisker boundaries are added to the usual boundary regions outside the horizon,
we saw that we arrive at a Lorentzian torus. Because of the existence of circular time in the
whisker boundaries, the CFT path integral does not have a canonical quantum mechanical
interpretation, in that we cannot simply specify any initial state in a Hilbert space and let
it evolve. Instead the path integral gives us an entire quantum spacetime which we can ask
questions of, in the form of correlations of Hermitian observables. In this sense, it has the
form of a kind of wavefunction of the Universe.
Alternatively, we can think of our results as simply demonstrating that the extended
black hole is a robust emergent phenomenon within a (single) hot CFT. For instance,
we saw in subsection 7.1 that with sources restricted to being outside the horizon, in
either exterior region L or R, our trace formula reduces to (7.1), which is equivalent to
the standard thermofield description, (7.3), (7.4). Local sources in L can be thought of as
specific non-local sources in R, so that there is a single CFT in a thermal heat bath,
Z[JL,R] = tr neH eH ULeH URo
This is just a re-writing of the thermofield description as a thermal trace in a single CFT,
rather than pure quantum mechanical evolution in two copies of the CFT. To describe
observables in L, we see we have to take standard observables and smear them between
eH and eH . In other words, local L observables are secretly just non-local observables
in R. In this view there is only the R CFT in a heat bath, and the L is an emergent
description to track certain non-local correlators. This is related to the discussion of the
emergence of doubling of CFTs in subsection 5.1 of [55]. Now, the results of our paper, in
particular the last line of (7.5), has shown that the UF,P probes of the inside-horizon F, P
regions can be thought of as emerging from non-local probes in the outside-horizon R
and L regions, arising from smearing standard R, L observables between eP and eP .
Putting all these observations together, we can think of probes anywhere in the extended
black hole spacetime as emerging from non-local correlators in a single CFT with thermal
density-matrix: eq. (6.10) can be re-expressed as
Z[JL,R,F,P ] = tr neH heH (eP UP eP )ULeH i (eP UF eP )URo
Non-local correlators in the thermal density matrix project the extended black hole,
including the singularity. This follows from our results. In this way, there is a modest
landscape of regimes of the gravitational dual, connected by horizons. Possibly other
non-local operators, not of the forms above, may project other parts of the landscape of
the quantum gravity dual.
We are tremendously indebted to Ted Jacobson for generously sharing his insights, advice
and criticism, and extensive knowledge of the literature. We are also grateful to Dieter
Brill, Juan Maldacena, Stephen Shenker and Eva Silverstein for enlightening discussions
and suggestions. We thank Nima Arkani-Hamed for alerting us to [55]. This research
was supported by the Maryland Center for Fundamental Physics, as well as by NSF grant
PHY-0968854 and by NSF grant PHY-1315155.
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